Question

Given below are two statements, one is labelled as Assertion (A) and the other is labelled as Reason (R): 
(A) A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. 
(R) The mass of the pendulum remains unchanged at Earth and the other planet. 
In light of the above statements, choose the correct answer from the options given below:

• A. (A) is false, but (R) is true.
• B. Both (A) and (R) are true and (R) is the correct explanation of (A).
• C. (A) is true but (R) is false.
• D. Both (A) and (R) are true, but (R) is NOT the correct explanation of (A).

Hint:

The time period of a simple pendulum depends on the acceleration due to gravity \( g \). Gravity is determined by the mass and radius of the planet.

Answer 1

The Correct Option is C

This problem analyzes a simple pendulum's behavior under varying gravitational acceleration. The formula for the time period \( T \) of a simple pendulum is given by \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) represents the pendulum's length and \( g \) is the acceleration due to gravity.

On Earth, the gravitational acceleration \( g \) is calculated as \( \frac{GM_e}{R_e^2} \), with \( G \) being the gravitational constant, \( M_e \) the Earth's mass, and \( R_e \) the Earth's radius.

For a planet with a mass of \( 4M_e \) and a radius of \( 2R_e \), the gravitational acceleration \( g' \) is derived as follows:

\( g' = \frac{G \cdot 4M_e}{(2R_e)^2} = \frac{4GM_e}{4R_e^2} = \frac{GM_e}{R_e^2} = g \)

Consequently, the time period \( T' \) on this planet is:

\( T' = 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{g}} = T \)

This demonstrates that the time period remains identical on both Earth and the planet, substantiating assertion (A).

Now, examining reason (R), which states that the pendulum's mass is constant in both locations. Although true, the pendulum's mass does not influence its time period \( T \), as indicated by the formula. Therefore, (R) does not provide an explanation for (A).

The correct conclusion is: (A) is true but (R) is false.